I tried to use this:
https://{CompanyTenant}.attask-ondemand.com/attask/api/task/search?{FieldName}={FieldValue}&apiKey={APIKey}&projectID={projectID}
Get Error!
my point is to find a {taskID} that has a given {FieldValue} on the specific {FieldName}.
Is FieldName a custom field or a system field? If it's the latter, that query should work, but if it's the former, you'll need to search for DE:{FieldName} as it's a Data Extension field.